Flat groups of automorphisms of totally disconnected, locally compact groups

TL;DR

This paper studies flat groups of automorphisms of totally disconnected locally compact groups, revealing their structure, properties, and analogies with Lie groups and buildings, along with new proofs of key results.

Contribution

It characterizes flat automorphism groups, their ranks, and structural features, providing new proofs and extending understanding of their properties in the context of totally disconnected groups.

Findings

Flat groups have a free abelian quotient of automorphisms.

Singly generated flat groups have rank 0 or 1.

Higher rank flat groups relate to Lie groups over local fields and buildings.

Abstract

A group, $\fl{H}$, of automorphisms of a totally disconnected locally compact group, $G$, is flat if there is a compact open $U\leq G$ such that the index $[\alpha(U):U\cap \alpha(U)]$ is mininimized for every $\alpha\in\fl{H}$. The stabilizer of $U$ in $\fl{H}$ is a normal subgroup, $\fl{H}_u$; the quotient $\fl{H}/\fl{H}_u$ is a free abelian group; and the rank of $\fl{H}$ is the rank of this free abelian group. Each singly generated group $\langle\alpha\rangle$ is flat and has rank either $0$ or $1$. Higher rank groups may be seen in Lie groups over local fields and automorphism groups of buildings. Flat groups of automorphisms exhibit many of the features of these special examples, including analogues of roots and a factoring of $U$ into analogues of root subgroups. New proofs of improved versions of these results are presented here.